higher K-groups
K_0, K_1, and K_2 were discovered one at a time, each by a different hand-built construction, and for a while it was unclear they belonged to a single infinite family. Quillen's breakthrough was to define all K-groups at once as the homotopy groups of a single space attached to a ring or category. Once you have such a space, its higher homotopy groups simply are the higher K-groups, and the earlier ad hoc definitions fall out as the bottom rungs of a uniform ladder.
There are two equivalent constructions. The plus construction: take the classifying space BGL(R), apply Quillen's plus construction with respect to the perfect subgroup E(R) to get BGL(R)^+, and define K_n(R) = pi_n(BGL(R)^+) for n at least 1, together with K_0(R) defined separately as the Grothendieck group. The Q-construction: for an exact category A, form the Quillen Q-category QA, take its classifying space BQA, and set K_n(A) = pi_{n+1}(BQA, 0); these agree with the plus-construction groups for A the projective modules. Both make K-theory a sequence of functors K_n.
The higher K-groups are genuinely deep and largely uncomputed. Quillen himself computed them for finite fields: K_{2i-1}(F_q) = Z/(q^i - 1)Z and K_{2i}(F_q) = 0 for i at least 1. For the integers the groups K_n(Z) are connected to values of the Riemann zeta function and the Vandiver conjecture, and many remain open. Higher K-theory satisfies powerful structural theorems — additivity, localization, dévissage, resolution — that make it computable in favorable cases despite the difficulty in general.
For the field F_4 with 4 elements, K_1(F_4) = F_4^* = Z/3Z, K_2(F_4) = 0, K_3(F_4) = Z/15Z (since q^2 - 1 = 15), and K_4(F_4) = 0. The pattern K_{2i-1} = Z/(4^i - 1)Z, K_{2i} = 0 continues for all i at least 1.
Quillen's computation of the K-groups of a finite field.
The two constructions agreeing is a theorem, not a definition; Quillen proved the plus and Q constructions give the same groups for projective modules. Later models — Waldhausen's S-dot construction, spectra, and motivic/higher-categorical approaches — extend the reach to settings (like categories with cofibrations) where the original constructions do not directly apply.